The proof follows from [1]}.

Notice that the vertical space is a subspace of \(T_{\bar{{\cal X}}}\overline{{\cal M}}_{\mathbf {r}}\) . The horizontal space, denoted by \({\cal H}_{\bar{{\cal X}}}\) , is any subspace of \(T_{\bar{{\cal X}}}\overline{{\cal M}}_{\mathbf {r}}\)  that is complementary to \({\cal V}_{\bar{{\cal X}}}\) . For the quotient manifold \(\overline{{\cal M}}_{\mathbf {r}}/{\cal G}\)  (REF ), any
element \(\bar{\xi }\in T_{\bar{{\cal X}}}\overline{{\cal M}}_{\mathbf {r}}\)  satisfying \(\mathrm {D}\pi \left(\bar{{\cal X}}\right)\left[\bar{\xi }\right]= \xi _{\left[\bar{{\cal X}}\right]}\)  can be seen as a representation of \(\xi _{\left[\bar{{\cal X}}\right]}\)  where \(\xi _{\left[\bar{{\cal X}}\right]}\in T_{\left[\bar{{\cal X}} \right]}\overline{{\cal M}}_{\mathbf {r}}/{\cal G}\) . Since the kernel of \(\mathrm {D}\pi \left(\bar{{\cal X}}\right): T_{\bar{{\cal X}}}\overline{{\cal M}}_{\mathbf {r}}\rightarrow T_{\left[\bar{{\cal X}} \right]}\overline{{\cal M}}_{\mathbf {r}}/{\cal G}\)  is the vertical space \({\cal V}_{\bar{{\cal X}}}\) , there are infinitely many representations of \(\xi _{\left[\bar{{\cal X}}\right]}\)  in \(T_{\bar{{\cal X}}}\overline{{\cal M}}_{\mathbf {r}}\) . Nevertheless,
one can find a unique representation of \(\xi _{\left[\bar{{\cal X}}\right]}\)  in horizontal space \({\cal H}_{\bar{{\cal X}}}\) . The tangent vector \(\bar{\xi }\in {\cal H}_{\bar{{\cal X}}}\)  satisfying \(\mathrm {D}\pi \left(\bar{{\cal X}}\right)\left[\bar{\xi }\right]= \xi _{\left[\bar{{\cal X}}\right]}\)  is called the horizontal lift of \(\xi _{\left[\bar{{\cal X}}\right]}\)  at \(\bar{{\cal X}}\) . Throughout this paper, the horizontal lift of \(\xi _{\left[\bar{{\cal X}}\right]}\in T_{\left[\bar{{\cal X}}\right]}\overline{{\cal M}}_{\mathbf {r}}\)  at \(\bar{{\cal X}}\)  is denoted by \(\bar{\xi }_{\bar{{\cal X}}}\) .
